It is utter nonsense to say that sqrt 2is irrational, because this presupposes that it exists, as a number or distance. The truth is that there is no such number or distance. What does exist is the symbol, which is just shorthand for an ideal object x that satisfies x2 = 2.
Now what the hell does that mean? A rational number is one that can be expressed as a fraction a/b where both a and b are integers and b is not 0. An irrational number is one that cannot be expressed in this way. By the celebrated theorem of Pythagoras, a right triangle with sides of 1 unit in length will have an hypotenuse with length = the square root of 2. This is an irrational number. But this irrational number measures a quite definite length both in the physical world and in the ideal world. How can this number not exist? It is inept to speak of a symbol as shorthand for an ideal object since, if x is shorthand for y, then both are linguistic items. For example 'POTUS' is shorthand for 'president of the United States.' But 'POTUS' is not shorthand for Obama. 'POTUS' refers to Obama. Zeilberger appears to be falling into use/mention confusion. If the symbol for the sqrt of 2 refers to an ideal object, then said object is a number that does exist. And in that case Zeilberger is contradicting himself.
What's more, it seems that from Zeilberger's own example one can squeeze out an argument for actual infinity. We note first that the decimal expansion of the the sqrt of 2 is nonterminating: 1.4142136 . . . . We note second that the length of the hypotenuse is quite definite and determinate. This seems to suggest that the decimal expansion must be actually infinite. Otherwise, how could the length of the hypotenuse be definite?
As an ultrafinitist, however, Zeilberger denies both actual and potential infinity:
. . . the philosophy that I am advocating here is calledultrafinitism. If I understand it correctly, the ultrafinitists deny the existence of any infinite, not [sic] even the potential infinity, but their motivation is `naturalistic', i.e. they believe in a `fade-out' phenomenon when you keep counting. [. . .]
So I deny even the existence of the Peano axiom that every integer has a successor.
As I said, this is wild stuff. He may be competent as a mathematician; I am not competent to pronounce upon that question. But he appears to be an inept philosopher of mathematics. But this is not surprising. It is not unusual for competent scientists and mathematicians to be incapable of talking coherently about what they are doing when they pursue their subjects. Poking around his website, I find more ranting and raving than serious argument.
The ComBox is open if someone can clue us into the mysteries of ultrafinitism. There is also some finitist Russian cat, a Soviet dissident to boot, name of Esenin-Volpin, who Michael Dummett refers to in his essay on Wang's Paradox, but Dummett provides no reference. Is ultrafinitism the same as strict finitism?